Problem: A circle with radius $5$ has a sector with a $216^\circ$ central angle. What is the area of the sector? ${25\pi}$ $\color{#9D38BD}{216^\circ}$ ${15\pi}$ ${5}$
Answer: First, calculate the area of the whole circle. Then the area of the sector is some fraction of the whole circle's area. $A_c = \pi r^2$ $A_c = \pi (5)^2$ $A_c = 25\pi$ The ratio between the sector's central angle $\theta$ and $360^\circ$ is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}$ $\dfrac{216^\circ}{360^\circ} = \dfrac{A_s}{25\pi}$ $\dfrac{3}{5} = \dfrac{A_s}{25\pi}$ $\dfrac{3}{5} \times 25\pi = A_s$ $15\pi = A_s$